| Autor: |
Exner, Pavel, Manko, Stepan S. |
| Rok vydání: |
2013 |
| Předmět: |
|
| Zdroj: |
J. Phys. A: Math. Theor. 46 (2013), 345202 (17pp) |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.1088/1751-8113/46/34/345202 |
| Popis: |
We investigate the limit properties of a family of Schr\"odinger operators of the form $H_\varepsilon= -\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ \frac{\lambda(\varepsilon)}{\varepsilon^2}Q \big(\frac{x}{\varepsilon}\big)$ acting on $n$-edge star graphs with Kirchhoff conditions imposed at the vertex. The real-valued potential $Q$ is supposed to have compact support and $\lambda(\cdot)$ to be analytic around $\varepsilon=0$ with $\lambda(0)=1$. We show that if the operator has a zero-energy resonance of order $m$ for $\varepsilon=1$ and $\lambda(1)=1$, in the limit $\varepsilon\to 0$ one obtains the Laplacian with a vertex coupling depending on $1+\frac12 m(2n-m+1)$ parameters. We prove the norm-resolvent convergence as well as the convergence of the corresponding on-shell scattering matrices. The obtained vertex couplings are of scale-invariant type provided $\lambda'(0)=0$; otherwise the scattering matrix depends on energy and the scaled potential becomes asymptotically opaque in the low-energy limit. |
| Databáze: |
arXiv |
| Externí odkaz: |
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